Combinatorial game theory

Combinatorial game theory is one of John Horton Conway c.1970 -founded branch of mathematics that deals with a special class of two-person games.

The properties of these games are:

• No coincidence influence.
• There is no for a single player hidden information (like playing cards ). that is, there is perfect information before.
• Pulled alternately.
• The winner is the player who manages to make the last train.
• Each game ends after a finite number of moves.

Such games, which include Nim and ( usually after minor transformations ) Go and Chess, open particularly interesting opportunities for mathematical analysis if they decompose into components for which there is no mutual influence of the possible moves. Examples are Nim - heap and some late playoff positions in Go; in chess, some Zugzwang positions with pawn endings can be so interpreted. Assembling positions is also known as addition.

The mathematical meaning of combinatorial game theory results from the fact that the games are a subclass can be interpreted as numbers. In this case, both whole and real and even transfinite (ie infinitely large and infinitely small ) numbers can be constructed, the entirety of which is also called surreal numbers. Conversely, the games of combinatorial game theory appear as a generalization of surreal numbers.

Who can win a targeted?

In terms of good strategies certainly achievable profit prospects each position belongs to exactly one of the following four classes:

• The first train kicker has a winning strategy, regardless of the style of play of his opponent secures him a profit.
• The subsequently immigrating player has a winning strategy.
• Regardless of who performs the first train, player 1 has, usually referred to as a left or white, has a winning strategy.
• Player 2, usually referred to as law or black, has a winning strategy.

A major component of combinatorial game theory is the so-called temperature theory. This makes it possible to approximeren the earnings outlook of a game from data of the individual components.

Special case: Neutral Games

Games where the properties listed above, the possible moves for both players are always identical in addition, hot -neutral games ( the original English term is sometimes also with objectively impartial translated). In relation to the earnings outlook includes any position of a neutral game to one of the two first classes of the above list.

A complete analysis of a neutral game is always possible because each position is assigned an equivalent, consisting of only a heap position of the normal Nim game. This possibility of reduction was independent of each other in 1935 by Roland Sprague and 1940 of Grundy found and is therefore also referred to as a set of Sprague- Grundy. Approaches to the reduction had already discussed the world chess champion and mathematician Emanuel Lasker.

The size of the nim cluster that is associated with a position of a neutral game is referred to as the Grundy number. The Grundy number one position can be relatively simple recursive, ie be calculated from the Grundy numbers of reachable positions in a train: it is equal to the smallest natural number that is not Grundy number of a successor position.

Grundy numbers have the following two properties:

• The attractive player has a winning strategy if and only about if the Grundy number is not 0.
• The Grundy number of a sum, that is, a composite of component position is equal to the XOR sum of the Grundy numbers of its components, which simplifies the computation complexity moderately critical in such cases.

The two properties generalize the winning strategy found in 1901 by Charles Leonard Bouton for the Nim game, after which you should draw as always, that the XOR sum of the heap sizes is equal to 0.

Example: Artificial finals at Go

Even if Go is not actually a game in which the last withdrawing player wins, the usual point ratings of the final match of Go can be transformed into corresponding rules of the game, which fulfill the conditions of combinatorial game theory. However, there is also an alternative approach based on Milnor (1953), Hanner (1959) and Berlekamp (1996 ) goes back. The point ratings and their properties are examined with respect to the components of a ( final ) position directly.